How to make a displacement map from existing 3d geometry
The best way I know is to use Baking.
You must have your plane (or whatever the low-res object is) UV mapped with a new image assigned to all the faces you want to bake to (select faces in edit mode, select/reselect the image in the Image Editor).
Then in the Render settings, at the bottom is the bake panel.

Normalized allows you to specify a maximum distance
Selected to Active is important too, enable it to ensure the displacement is baked from all selected objects to the active/last selected object.
Example:

(Note that baking doesn't play nice with overlapping geometry - in this example the underneath of the sphere and torus was deleted, and the sphere moved up to ensure it doesn't collide with the torus. Not doing this will produce artifacts.)